Simplify (a-b)/(a^2-ab+b^2) + (a+b)/(a^2+ab+b^2)

Hey guys! Today, we're diving into a cool algebraic simplification problem. We're going to break down this seemingly complex expression step by step, so it becomes super clear and easy to understand. Let's get started!

Understanding the Problem

Okay, so we have this expression:

(a-b)/(a^2-ab+b2) + (a+b)/(a^2+ab+b2) - (2a^3)/(a4-a^2b2+b^4)

At first glance, it might look a bit intimidating, but don't worry! We're going to simplify it piece by piece. The key here is to recognize patterns and use algebraic identities to our advantage. We need to simplify this expression involving fractions with polynomials. To do this, we'll focus on finding common denominators and combining like terms. The goal is to make the expression as simple as possible. This involves several steps, including factoring, finding common denominators, combining fractions, and simplifying the result. Simplifying algebraic expressions like this is a fundamental skill in mathematics, with applications in various fields, including calculus, physics, and engineering. Mastering these techniques helps in solving complex equations and understanding mathematical relationships more clearly. By the end of this guide, you'll be able to tackle similar problems with confidence. We will go through each step meticulously, ensuring you grasp the underlying concepts and can apply them to other problems. So, stick with me, and let's make math a little less scary and a lot more fun!

Initial Observations and Strategy

Before we jump into the nitty-gritty, let's take a moment to observe the expression. We have three fractions here. Notice that the denominators have some similarities. The first two denominators, a^2 - ab + b^2 and a^2 + ab + b^2, look like they could be part of a larger pattern. The third denominator, a^4 - a^2b2 + b^4, seems more complex, but we'll see how it relates to the other two soon. Our main strategy here is to combine these fractions. To do that, we need a common denominator. Finding this common denominator will be our first big step. Once we have a common denominator, we can add and subtract the numerators. After combining the fractions, we'll look for opportunities to simplify further. This might involve factoring, canceling out common terms, or using algebraic identities. Remember, the goal is to make the expression as simple as possible. We'll take it one step at a time, making sure we understand each step before moving on. This methodical approach will help us avoid mistakes and keep things clear. So, let's roll up our sleeves and get started!

Step-by-Step Simplification

Alright, let's break this down into manageable steps.

Step 1: Finding a Common Denominator

Okay, so the first thing we need to do is find a common denominator for our three fractions. Remember our expression:

(a-b)/(a^2-ab+b2) + (a+b)/(a^2+ab+b2) - (2a^3)/(a4-a^2b2+b^4)

The denominators are a^2 - ab + b^2, a^2 + ab + b^2, and a^4 - a^2b2 + b^4. To find the common denominator, we need to see how these expressions relate to each other. Let's focus on the third denominator, a^4 - a^2b2 + b^4. This might look familiar if you've worked with algebraic identities before. It's actually a variation of a difference of squares pattern. Specifically, it can be seen as (a^2 + b²⁾2 - (ab)^2, which is similar to x^2 - y^2. But a more helpful way to think about it is as a product of the first two denominators. If we multiply (a^2 - ab + b^2) and (a^2 + ab + b^2), we get:

(a^2 - ab + b^2)(a2 + ab + b^2) = a^4 + a^3b + a^2b2 - a^3b - a^2b2 - ab^3 + a^2b2 + ab^3 + b^4 = a^4 + a^2b2 + b^4

Oops! It seems we made a slight error in our initial observation. The product is actually a^4 + a^2b2 + b^4, not a^4 - a^2b2 + b^4. But don't worry, this is a common mistake, and we'll adjust our approach accordingly. However, this gives us a crucial insight: The correct factorization would involve a small tweak to the middle term. The correct expression a^4 - a^2b2 + b^4 can be thought of as (a^2 + b²⁾2 - 3a^2b2, which doesn't directly factor in a way that helps us. So, instead of forcing a factorization, let's recognize that the least common denominator (LCD) will be the product of all unique factors present in the denominators. In this case, since we can't neatly factor a^4 - a^2b2 + b^4 into the other two, the LCD will simply be the product: (a^2 - ab + b^2)(a2 + ab + b^2). Or, more conveniently, a^4 + a^2b2 + b^4 as we found earlier (Corrected).

Therefore, the common denominator we'll use is a^4 + a^2b2 + b^4. This is a key step, so make sure you're comfortable with how we found it. Now, let's move on to the next step, where we'll rewrite our fractions using this common denominator.

Step 2: Rewriting Fractions with the Common Denominator

Now that we have our common denominator, a^4 + a^2b2 + b^4, we need to rewrite each fraction with this denominator. Let's start with the first fraction:

(a-b)/(a^2-ab+b2)

To get the denominator a^4 + a^2b2 + b^4, we need to multiply both the numerator and the denominator by (a^2 + ab + b^2):

[(a-b)(a^2 + ab + b^2)] / [(a^2-ab+b2)(a^2 + ab + b^2)]

Let's expand the numerator:

(a-b)(a^2 + ab + b^2) = a^3 + a^2b + ab^2 - a^2b - ab^2 - b^3 = a^3 - b^3

So, the first fraction becomes:

(a^3 - b^3) / (a^4 + a^2b2 + b^4)

Now, let's move on to the second fraction:

(a+b)/(a^2+ab+b2)

To get the common denominator, we multiply both the numerator and denominator by (a^2 - ab + b^2):

[(a+b)(a^2 - ab + b^2)] / [(a^2+ab+b2)(a^2 - ab + b^2)]

Expand the numerator:

(a+b)(a^2 - ab + b^2) = a^3 - a^2b + ab^2 + a^2b - ab^2 + b^3 = a^3 + b^3

So, the second fraction becomes:

(a^3 + b^3) / (a^4 + a^2b2 + b^4)

The third fraction already has the common denominator:

(2a^3) / (a^4 - a^2b2 + b^4), there was a typo on the previous calculation so we need to fix it.

Let's rewrite a^4 - a^2b2 + b^4 to the common denominator a^4 + a^2b2 + b^4, we need to multiply the numerator and denominator by (a^4 + a^2b2 + b^4)/(a4 - a^2b2 + b^4). However, to keep the original value we must consider this fraction will remain the same.

So, the third fraction remains:

(2a^3) / (a^4 - a^2b2 + b^4)

Now, we have all three fractions with a common denominator (or adjusted to have one). In the next step, we'll combine these fractions and see what simplifications we can make.

Step 3: Combining the Fractions

Great! We've got all our fractions with a common denominator (or correctly represented). Now, let's combine them. Our expression looks like this:

(a^3 - b^3) / (a^4 + a^2b2 + b^4) + (a^3 + b^3) / (a^4 + a^2b2 + b^4) - (2a^3) / (a^4 - a^2b2 + b^4)

Combining the first two fractions is straightforward since they have the same denominator:

[(a^3 - b^3) + (a^3 + b^3)] / (a^4 + a^2b2 + b^4) = (2a^3) / (a^4 + a^2b2 + b^4)

Now, we have:

(2a^3) / (a^4 + a^2b2 + b^4) - (2a^3) / (a^4 - a^2b2 + b^4)

To combine these two fractions, we need a common denominator, which will be the product of the two denominators:

[(2a^3)(a4 - a^2b2 + b^4) - (2a^3)(a4 + a^2b2 + b^4)] / [(a^4 + a^2b2 + b^4)(a4 - a^2b2 + b^4)]

Let's expand the numerator:

(2a^7 - 2a^5b2 + 2a^3b4) - (2a^7 + 2a^5b2 + 2a^3b4) = 2a^7 - 2a^5b2 + 2a^3b4 - 2a^7 - 2a^5b2 - 2a^3b4 = -4a^5b2

Now, let's expand the denominator. This is a bit trickier, but we can do it systematically:

(a^4 + a^2b2 + b^4)(a4 - a^2b2 + b^4) = a^8 - a^6b2 + a^4b4 + a^6b2 - a^4b4 + a^2b6 + a^4b4 - a^2b6 + b^8 = a^8 + a^4b4 + b^8

So, our expression becomes:

(-4a^5b2) / (a^8 + a^4b4 + b^8)

This looks much simpler already! In the next step, we'll see if we can simplify this further.

Step 4: Final Simplification

Okay, so we've reached a point where our expression looks like this:

(-4a^5b2) / (a^8 + a^4b4 + b^8)

Now, let's see if we can simplify this fraction any further. We need to look for common factors in the numerator and the denominator. The numerator is -4a^5b2. The denominator is a^8 + a^4b4 + b^8. At first glance, it might not be obvious if there are any common factors. However, we can try to express the denominator in a different form to see if anything cancels out. The denominator looks a bit like a quadratic expression if we think of a^4 as a single variable. But it doesn't factor easily in the traditional sense. Let's try another approach. We can try to complete the square in the denominator. Notice that if we had a^8 + 2a^4b4 + b^8, it would be a perfect square: (a^4 + b⁴⁾2. Our expression is close, but it's missing an a^4b4 term. We can rewrite the denominator as:

a^8 + 2a^4b4 + b^8 - a^4b4 = (a^4 + b⁴⁾2 - (a^2b2)^2

Now, we have a difference of squares! We can factor this as:

[(a^4 + b^4) + a^2b2][(a^4 + b^4) - a^2b2] = (a^4 + a^2b2 + b^4)(a4 - a^2b2 + b^4)

So, our expression now looks like:

(-4a^5b2) / [(a^4 + a^2b2 + b^4)(a4 - a^2b2 + b^4)]

Unfortunately, this factorization doesn't directly lead to any cancellations with the numerator. It seems we've explored all the common algebraic techniques, and we're at the simplest form of the expression. Therefore, the final simplified expression is:

(-4a^5b2) / (a^8 + a^4b4 + b^8)

Conclusion

Alright, guys! We made it! We successfully simplified the expression:

(a-b)/(a^2-ab+b2) + (a+b)/(a^2+ab+b2) - (2a^3)/(a4-a^2b2+b^4)

After several steps of finding common denominators, combining fractions, and simplifying, we arrived at the final answer:

(-4a^5b2) / (a^8 + a^4b4 + b^8)

This problem was a great exercise in algebraic manipulation. We used key techniques like finding common denominators, expanding products, and looking for opportunities to factor. Remember, the key to simplifying complex expressions is to take it one step at a time. Don't be afraid to make mistakes – they're part of the learning process! By carefully working through each step, we can break down even the most intimidating-looking problems into manageable pieces. I hope you found this walkthrough helpful! Keep practicing, and you'll become a pro at simplifying algebraic expressions in no time. If you have any questions or want to try another problem, just let me know. Happy simplifying!